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Adaptive Methods for the Poisson

Equation in Complex Geometry

We are working on fast, adaptive methods for the solution of the Poisson equation

Δ U = f

in complex geometry, subject to linear boundary conditions in interior or exterior domains. In simple geometries (circular or rectangular domains) with regular grids, there are well-known fast direct solvers based on the fast Fourier transform (FFT) that are well-suited to the task. In practical problems, however, involving complex geometries, highly inhomogeneous source distributions (f ), or both, there has been a lot of effort directed at developing alternative approaches. Most currently available solvers rely on iterative techniques using multigrid, domain decomposition, or some other preconditioning strategy. Although there has been significant progress in this direction, the available solvers compare unfavorably with fast direct solvers in terms of work per gridpoint. We are developing methods which are direct, high-order accurate, insensitive to the degree of adaptive mesh refinement, and accelerated by the FMM. Our goal is to produced solvers that are competitive (or nearly competitive) with standard fast solvers in terms of work per gridpoint . Technical references: